Question

Let $R$ be a relation defined on the set $Z$ of all integers and $xRy$ when $x+2y$ is divisible by $3$. Then

• A. $R$ is not transitive
• B. $R$ is symmetric only
• C. $R$ is an equivalence relation
• D. $R$ is not an equivalence relation

Answer 1

The correct option is C $R$ is an equivalence relation

$aRa$ is positive as $a+2a=3a$ is divisible by $3$

Hence reflexive

If $aRb$ is positive $a+2b$ is divisible by $3$

Hence $3a+3b-(2a+b)$ is divisible by $3$

$\Rightarrow 2a+b$ is divisible by $3$

ie $aRb$ is positive

$\Rightarrow bRa$ is positive hence symmetric

$aRb,bRc$ is $a+2b,b+2c$ divisible by $3$

$\Rightarrow a+2b+b+2c$ divisible by $3$

$\Rightarrow a+3b+2c$ divisible by $3$

So $a+2c$ divisible by $3$ hence transitive

So it is an equivalence relation.